Assignment 13

  1. Use integration by substitution to solve the integral below:

\[ \int4e^{-7x}dx\] \[ u = -7x\] \[ du = -7dx\] \[ \frac{du}{-7} = dx\] First substitution

\[ \int4e^{u}\frac{du}{-7}\] \[ \int\frac{4}{-7}e^{u}du\] \[ \frac{4}{-7}\int e^{u}du\] \[ \frac{4}{-7} e^{u} + C\]

bring back the first substitution to get the final answer:

\[ \frac{4}{-7} e^{-7x} + C\]

  1. Biologists are treating a pond contaminated with bacteria. The level of contamination is changing at a rate of \(\frac{dN}{dt}=\frac{3150}{t4}−220\) bacteria per cubic centimeter per day, where t is the number of days since treatment began. Find a function N( t ) to estimate the level of contamination if the level after 1 day was 6530 bacteria per cubic centimeter.

we are solving dor Dn with variables t:

\[\frac{dN}{dt}=\frac{-3150}{t4}−220\]

\[dN=\frac{-3150}{t4}−220dt\] Integrate the two parts:

\[dN=\int\frac{-3150}{t4}−\int220dt\] \[N(t)=\int\frac{3150}{t4}−\int220dt\]

  1. Find the total area of the red rectangles in the figure below, where the equation of the line is f ( x ) = 2x - 9.

  2. Find the area of the region bounded by the graphs of the given equations. \(y = x^2 - 2x - 2, y = x + 2\)

  3. A beauty supply store expects to sell 110 flat irons during the next year. It costs $3.75 to store one flat iron for one year. There is a fixed cost of $8.25 for each order. Find the lot size and the number of orders per year that will minimize inventory costs.

I’m not sure how to approach this problem